Quantification in predicate logic and in English Lecture Two 24 October 2014

ACTL course, UCL, 2014 Dimitra Lazaridou-Chatzigoga (co-taught with Yasu Sudo)

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Last week: quick recap •  Intro to formal semantics • What is meaning? •  Truth conditions •  Compositional • Model-theoretic •  Function(al) application •  Quantificational expressions

From last week

•  Q: How do natural languages express quantification?

•  English vs. Predicate Logic (see Lecture 2) 1)  a. No boy has a cat.

b. ¬∃x[boy(x)∧∃y[cat(y)∧have(x,y)]] 2)  a. Every boy has a cat.

b. ∀x[boy(x) ⇒ ∃y[cat(y)∧have(x,y)]]

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Plan for today

• What is logic? •  Predicate logic (first-order logic) •  The universal quantifier •  The existential quantifier •  Defining other quantifiers •  Comparison with English

Review: what is logic? •  Attempts to discover, characterize, and

explain valid reasoning •  that is, elucidate patterns or schemas of

reasoning that are truth preserving, or never lead from truth to falsity

(in contrast with merely plausible ones)"

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Logic cont •  The attention of logic is further restricted to those inferences

that are formally valid 1.  a. The cat is black or it is ginger. Therefore, if it is not black, then it is ginger. b. Yasu is male. Therefore Yasu is human. •  Formal formal validity is due to structure rather than subject

matter: e.g. (1-a) vs. (1-b) •  In terms of language: due to meanings of functional or

closed class, rather than lexical vocabulary? •  Due to functional or closed class vocabulary"

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Logic and natural language semantics

•  Thus have some overlapping aims… •  In studying valid reasoning, logicians are (at

least in part) studying the truth conditional contribution of (certain) functional or closed class vocabulary,

• Which as discussed last week is an important part of giving a theory of meaning for natural languages."

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Logic and natural language semantics

though their aims also diverge somewhat. . . •  Logic has a normative dimension -a concern with how we

should reason; linguistics is solely focused on actual competencies.

•  Since reasoning happens at the level of complete thoughts, and since logic aims at something universal (rather than language particular), it abstracts away from certain details of (great) linguistic interest - logicians are not concerned directly with how the syntax of a natural language gets mapped to meaning.

•  Thus, logic has proceeded by developing artificial or ‘formal’ languages which abstract away from or collapse such details."

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Why study logic in linguistics/semantics?

•  In short: knowing logic is nearly necessary, both conceptually and practically to study semantics.

•  Key tools and concepts used in semantics come from logic, and are more easily introduced there in abstraction from linguistic complexity: e.g. treatments of quantification, binding, scope, etc.

•  Logical languages like predicate logic are “trade languages”: semanticists use them both to talk about meaning, and in stating their own theories."

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Formal and natural languages FORMAL LANGUAGES: •  Languages of logic, of

mathematics, of programming •  They are constructed by humans for

a clear and specific reason •  They are learned after hard work •  They are build in such a way, that

they avoid ambiguity and vagueness •  They have only declarative

sentences •  They can be used as metalanguages

NATURAL LANGUAGES: •  People use them naturally •  They are learned as first languages,

mother tongues •  They are apt for communicative

reasons •  They have not been build/constructed

by someone •  They are multidimensional, that is,

there is a historic evolution that indicates continuous natural change

•  They are ambiguous and vague •  They include declarative, interrogative,

imperative, performative, exclamatives sentences etc.

•  They can be used as metalanguages

Reasoning with Quantifiers •  Quantificational expressions (for example some, all, no) allow us

to move beyond particulars to generality, and therefore are central in reasoning.

Every man is mortal. Socrates is a man. Therefore: Socrates is mortal. •  This is a a valid argument. Why? •  Explaining why eluded logicians for centuries: •  1. propositional logic appears of no help, [see slide after next] •  2. because its validity depends on the internal structure of the

propositions involved, every is responsible

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Valid arguments •  Frege, who “invented” (discovered?) predicate logic, overcame

this problem by working out a system, according to which predicates are expressions with ‘open places’, which could be filled by arguments that function as ‘placeholders’

•  That way it became possible to analyze not only predicate-

argument structures like ‘Socrates is mortal’, but also quantified statements like ‘Every man is mortal’

•  This system is called predicate logic or first order logic: it is ‘first order’ because predication and quantification always range over individuals, not properties or relations, which are higher order entities"

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Diversion: propositional logic •  Atomic statements like p, q, r… •  Logical connectives: ¬, ∧, ∨, →, ↔ •  Truth tables for connectives

•  Dimitra does not play the oboe. ¬ p •  Dimitra plays the piano and Yasu plays the oboe. p ∧ q •  Dimitra plays the piano or Yasu plays the oboe. p ∨ q •  If Dimitra plays the piano, then Yasu plays the oboe. p → q •  Dimitra will go to the concert only if Yasu does. p ↔ q

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Predicate logic

•  First order logic •  Internal structure of propositions

Every man is mortal. Socrates is a man. Therefore: Socrates is mortal.

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Predicate logic: examples of WFFs

•  Hackney is cool. C(h) one-place predicate •  Westminster is not cool. ¬C(w) •  Dimitra emailed Yasu. Edy or E(d,y) two-place predicate •  Yasu emailed Dimitra. Eyd or E(y,d) •  All the above involve proper names as arguments. What about

quantifiers like everyone or nothing?

WWFs •  Everything is cool. ∀xCx ‘everything is such that it is cool’ or equivalently ∀yCy •  There exists a thing that is cool ∃xCx •  Nothing is cool ¬∃xCx

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Vocabulary

•  a, b, c … are individual constants •  x, y, z … are individual variables •  P, Q, R … are predicate constants •  = is a binary identity predicate •  the connectives of propositional logic: ¬, ∧, ∨, →, ↔

•  Quantifiers ∃ and ∀ •  Brackets ( ) NB: Individual constants and individual variables are terms

Syntax a.  If P is a one-place predicate and t is a term, then Pt is a wff b.  If R is a two-place predicate and t1 and t2 are terms,then

Rt1t2 is a wff* c.  If φ and ψ are (any) wffs, then so are ¬φ, (φ∧ψ), (φ ∨ ψ),

(φ → ψ), (φ ↔ ψ) d.  If φ is a wff and x is an individual variable, then ∀xφ and

∃xψ are wff e.  nothing else is a wff

* if R is an n-place predicate, and t1, t2, … tn are individual constants or variables, then Rt1t2…tn

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WWFs •  Are the following WFFs? •  Sx •  ∀xPy

•  They are WFFs, because the syntax allows for vacuous quantification

•  Connectives may combine formulas with or without quantifiers c. If φ and ψ are (any) wffs, then so are ¬φ, (φ∧ψ), (φ ∨ ψ),

(φ → ψ), (φ ↔ ψ) •  Any quantifier plus a variable x can be prefixed to a formula φ,

even when x does not occur in φ (d) d. If φ is a wff and x is an individual variable, then ∀xφ and ∃xψ are

wff

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•  Dimitra does not cycle. ¬ C(d) •  Dimitra does not cycle, but Yasu does. ¬ C(d) ∧ C(y) •  If Dimitra cycles, Yasu is happy. C(d) → H(y)

More translations

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Quantifiers •  Every musician arrived late. wrong: A(e) •  No musician arrived late. wrong: A(n) •  ‘Every musician’ and ‘no musician’ do not denote

specific individuals, thus we cannot use an individual constant as we did with proper names

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Quantifiers •  The first sentence does not ascribe the property of arriving late

to the individual called ‘every musician’. Rather, if Dimitra, Yasu and Klaus constitute the set of musicians, it says that it is true for each one of them that he or she arrived late

•  The second sentence does not ascribe the property of arriving late to the empty set of musicians. Rather, if Dimitra, Yasu and Klaus constitute the set of musicians, it says that none of them arrived late

•  Everybody, somebody, nobody etc. are non-referential NPs, they do not refer to individuals.

•  What is then their denotation? •  The meaning of everyone, somebody, nobody etc. could be

similar to that of ∀, ∃ and the combination with the logical connectives

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Understanding quantifiers and variables Some comments are needed about WFFs like ∀xCx [everything is

cool] •  Variables appear as arguments of predicates, but unlike

constants their denotation is not fixed once and for all by the semantics.

•  Rather, they are placeholders whose denotation or value can vary:

an open formula like Cx cannot be evaluated without further specification.

•  Quantifiers can complete open formulas, yielding truth-evaluable

ones. They do this, coarsely put, by specifying how to interpret the variable that they are prefixed to."

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Quantifiers 1.  ∀xφ is true iff φ is true for all (possible) values for x 2.  ∃xφ is true iff φ is true under some -at least one- value for x Example: 1.  ∀xCx is true iff Cx is true for all values of x i.e. iff Cx is true

no matter what x should stand for"

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The “duality” of ∀ and ∃ •  ∃ and ∀ are related by negation in a particular way, known as

being ‘duals’: 1. ¬∀xFx is equivalent to ∃x¬Fx 2. ¬∃xFx is equivalent to ∀x¬Fx 3. ∀xFx is equivalent to ¬∃x¬Fx 4. ∃xFx is equivalent to ¬∀x¬Fx

•  No student complained. •  ∀x(Sx → ¬Cx) or ¬∃x (Sx ∧ Cx)

•  Not every student was happy. •  ¬∀x(Sx → Hx) or ∃x (Sx ∧ ¬Hx)

‘at least one’ •  ∃xφ is true iff φ is true under some -at least one- value for x •  The above highlights the fact that ∃xφ means “one or more

things are φ".

•  Certainly then the natural language quantifier ‘at least one thing' can be adequately represented as ∃. What about ‘something’?

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Some •  Something is incorrect.

may seem to suggest, in many contexts, that not everything (or even, many things) is. However consider:

•  Something is incorrect, but not everything. •  Something is incorrect, and perhaps everything is. •  These suggest that there is at least a reading of ‘something' that

means ∃. •  the suggestion of ‘not all’ where it does arise, is naturally accounted

for as a kind of conversational implicature (owing to the fact that the speaker could have chosen the more informative - logically stronger – statement ‘everything is incorrect'. . . )

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Translating restricted (universal) quantifiers!

•  We talk about everything, but what about a sentence like ‘Every cat is happy’ that makes a claim only about cats?

•  Since ∀ ranges over everything there is, a trick is needed to

render non-cats irrelevant, in order to capture the truth conditions of the English sentence in predicate logic.

•  Here it is: ∀x(Cx → Hx) to be read: for every thing x, if x is a cat then x is happy

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‘Every cat is happy’ •  By the ‘informal’ semantic rule for ∀ repeated here: •  ∀xφ is true iff φ is true for all (possible) values for x •  ∀x(Cx → Hx) is true iff ▫  letting ‘x’ denote your mom, Cx → Hx is true and ▫  letting ‘x’ denote Greece, Cx → Hx is true and ▫  letting ‘x’ denote London, Cx → Hx is true and

‘Every cat is happy’

Cx Hx Cx → Hx

T T T

T F F

F T T

F F T

•  As the table shows, Cx → Hx is falsified only by values for x that are unhappy cats.

•  Thus, (by the rule above): •  If there are unhappy cats, ∀x(Cx → Hx) is false. •  If there are not, ∀x(Cx → Hx) is true. "

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‘Every cat is happy’ •  This appears to correspond to intuitions about the truth

conditions of ‘every cat is happy’, showing our translation to be correct. With one caveat.

•  Every friend of Mary is present. •  Suppose Mary has no friends. Is this true, or false? •  What about a translation according to the above procedure? •  ∀x(Fx → Px) •  We would render it inappropriate or nonsensical

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Quantifiers and connectives •  Every cat is happy. ∀x(Cx → Hx) Wrong: ∀x(Cx ∧ Hx) •  Some cats slept. ∃x(Cx ∧ Sx) Wrong: ∃x(Cx → Sx) •  The connectives cannot be exchanged. Why? •  ∀x(Cx ∧ Hx) means that everything in the world is a happy cat •  ∃x(Cx → Sx) means that there is something in the world that if it is a

cat, then it is slept •  The truth conditions for implication specify that the formula comes

out true when the antecedent is false. Then ∃x(Cx → Sx) would be T if there is any individual that is not a cat, irrespective of whether there is anyone sleeping

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Scope in logic and English •  Descriptively, the term (semantic) scope is used to refer to the “semantic domain” of a word or expression: that part of the sentence that its meaning affects of has influence over.

•  Dimitra listened to music and cooked pasta or went to the

movies •  a. (Ld ∧ Cd) ∨ Gd •  b. Ld ∧ (Cd ∨ Gd) •  In predicate logic scope is explicitly determined by syntactic

structure. Removing any of the brackets in (a) or (b) does not yield an ambiguity; it obviates well-formedness.

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Scope and connectives a.  The scope of ∧,∨, or → in a WFF φ is (includes) the WFF that

appears to its left, and the WFF that appears to its right. b.  The scope of ¬ is the the WFF that appears to its right. •  While scope in the English sentence above may not be

transparent, it seems that once it is assigned a structure, scope and hence meaning are fixed.

•  In other English sentences, to which we turn in the coming

weeks, it is not clear that scope is fixed by assigning (observable) structure, raising interesting questions.

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Scope of quantifiers •  Like connectives, the domain of semantic influence of

quantifiers, their scope, is determined explicitly by the syntax: •  For any wff ∀xφ or ∃xφ, the scope of the quantifier, ∀x / ∃x, is φ, and any variable, quantifier, connective, or wff contained in φ is in the scope of ∀x / ∃x •  a. Ey ∧ ∀yCy •  b. ∀y¬Cy •  c. ¬∀yCy

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Bound and free variables [the definition of scope is necessary for the following definition:] •  A variable x is bound if it is in the scope of some occurrence of ∀x or ∃x, and otherwise is free;

•  x is free within φ if there is no occurrence of ∀x or ∃x, within φ that x is in the scope of

•  a. Ey ∧ ∀yCy •  b. ∀y(Ey∧Cy) •  c. ∀y(Ey∧Cx) •  A formula with no free variables is sometimes called a (closed)

sentence, and a formula with with free variables an (open) sentence.

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Binding •  Binding is hence a relation between a prefixed

quantifier and an occurrence of a variable •  Px •  ∀yFy ∨ Gy •  ∃x((Fx ∧ Gx) ∨ Fy) •  ∃yFy ∧ Gy •  Blue variables are bound, green ones is free

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Bound and free variables •  We could think of the difference between bound and free

variables as corresponding to the following contrast in a natural language like English:

•  Every student sang and danced. •  Every student sang and she danced.

•  The pronoun she in the second sentence cannot be dependent for its interpretation on the quantified NP every student, but should rather get its reference from somewhere else

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Rules of inference •  Descriptions of valid argument schemas •  The familiar ones from propositional logic + UI and EG •  Universal Instantiation •  ∀x Px Pc Every man is mortal. Socrates is a man. Therefore: Socrates is mortal. •  Existential Generalization •  Pc_ ∃xPx •  Dimitra played the oboe. Someone played the oboe.

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Semantics of predicate logic

• model •  valuation function •  values for constants and predicates •  assignment function that maps variables to

objects in the domain of discourse

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Defining other quantifiers

• No student is happy. ¬∃x(Sx ∧ Hx) • Not every student is happy. ¬∀x (Sx → Hx) NB: We translated using ¬, which is sentential and fortunately in

this case, it gives the right truth conditions-this is not always the case

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Using identity •  Someone loves someone.

Assume that the domain contains only people, and consider: 1. ∃x∃yLxy 2. ∃x∃y(Lxy ∧ ¬=xy) •  1 and 2 are not equivalent. 2, but not 1, says that someone

loves someone other than him/herself. This illustrates an important point about variables, i.e. there is nothing to stop variables denoting the same thing.

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Numerals •  Key: S: student, H: happy. 1. ∃x(Sx ∧ Hx) 2. ∃x∃y (Sx ∧ Hx) ∧ (Sy ∧ Hy) 3. ∃x∃y(((Sx ∧ Hx) ∧ (Sy ∧ Hy) ∧¬=xy

•  1 and 2 are equivalent; remember, there is nothing to stop two variables denoting the same thing. They both mean ‘a student is happy’ (i.e., ‘at least one student is happy').

•  3 is not the same, because in this case we have specified that x and y do not denote the same thing. 3 means ‘(at least) two students are happy'.

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“Modified” numerals •  Exactly one student is happy. 1. ∃x(Sx ∧ Hx) ∧ ¬∃x∃y (((Sx ∧ Hx) ∧ (Sy ∧ Hy)) ∧ ¬=xy 2. ∃x((Sx ∧ Hx) ∧ ∀y ((Sy ∧ Hy)) ∧ =xy))

•  1 and 2 are equivalent, and they are both predicate logic translations of the sentence above.

•  1 says ‘at least one student is happy, and it is not the case that at least two students are happy', while 2 says ‘there is at least one student who is happy and who is identical to every happy student’-notice that there cannot be more than one such student.

•  2 can also be paraphrased, somewhat less precisely, as ‘At least one student is happy, and every happy student is that student'.

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“Modified” numerals •  Less than two students are happy. •  At most two students are happy.

¬∃x∃y (((Sx ∧ Hx) ∧ (Sy ∧ Hy)) ∧ ¬=xy)

•  Very many (more) meanings that depend on counting can be

expressed in predicate logic (if not compactly). We wouldn't be able to do this without the = symbol.

•  As we will see next time, however, there is a limit.

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The definite article •  We can give a rough translation of the definite

article too:

•  The marathon gold-metallist is Ethiopian. •  ∃x((Gx ∧ ∀y ((Gy → =xy) ∧ Ex)) •  ‘There is exactly one thing that won gold, and that

thing is Ethiopian’

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Russell’s theory of descriptions

•  The King of France is bald.

•  ∃x(Kx ∧ ∀y ((Ky → =xy) ∧ Bx))

•  Existential commitment: there is a KoF •  Uniqueness requirement: there is only one

KoF •  He is bald

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Summary

•  Predicate logic (exercises to be distributed for next week)

•  Quantifiers ∀ and ∃ •  Other quantifiers •  Differences bt. Predicate logic and English:

v Constituents v Vacuous quantification v Restricted quantification v Quantification over an empty domain v Other kinds of quantifiers? See next week

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Other limits of first-order logic •  In natural language we talk about much more than properties of entities. •  Yasu is healthy. H(y) •  Running is healthy. H(r) •  Yasu has all the properties of Santa Claus. ∀P (P(s) → P(y) •  Yellow has something in common with orange. ∃P (P(Y) ∧ P(O))

•  H(r): ill-formed, because we can only predicate things of individuals in FOL •  ∀P (P(s) → P(y) ill-formed, because it expresses quantification over properties •  ∃P (P(Y) ∧ P(O)) ill-formed, because it expresses quantification over properties

of properties

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